polynomial fitting for COMPLEX data

A package which calls itself
"an industry-leading scientific graphing and data analysis software"
suggests breaking the samples into real and imaginary parts, and
fitting curves to each. Hmmmph. I guess it is not a common task
that they could be bothered coding.
Now surely, one can just set up the Vandermonde matrix, where the
elements are the sums of x, x squared, x cubed et cetera.
Or with complex data, would it end up as Hermitian (upper triangle
is conjugate of lower triangle)?

<mbjorn@y7mail.com> wrote:
>A package which calls itself
>"an industry-leading scientific graphing and data analysis software"
>suggests breaking the samples into real and imaginary parts, and
>fitting curves to each. Hmmmph. I guess it is not a common task
>that they could be bothered coding.
Are the abscissae complex, or are they real?
If real, I think doing this is the appropriate answer even if the range
is complex.
If complex, you can still do a Lagrangian interpolation but it
would take some working out of equations.
There are possibly also nuances depending on whether the abscissae
are uniformaly spaced.
Steve

0

spope33

11/12/2016 9:03:59 AM

On Sat, 12 Nov 2016 09:03:59 +0000, Steve Pope wrote:
> <mbjorn@y7mail.com> wrote:
>
>>A package which calls itself "an industry-leading scientific graphing
>>and data analysis software"
>>suggests breaking the samples into real and imaginary parts, and fitting
>>curves to each. Hmmmph. I guess it is not a common task that they could
>>be bothered coding.
>
> Are the abscissae complex, or are they real?
>
> If real, I think doing this is the appropriate answer even if the range
> is complex.
Yes, at least if you're looking for a least-squares fit. If you're
looking for a fit that minimizes some nonlinear cost criteria then the
real and complex solutions may well interact.
> If complex, you can still do a Lagrangian interpolation but it would
> take some working out of equations.
>
> There are possibly also nuances depending on whether the abscissae are
> uniformaly spaced.
>
> Steve
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
I'm looking for work -- see my website!

0

Tim

11/13/2016 10:46:11 PM

On 14/11/2016 01:46, Tim Wescott wrote:
> On Sat, 12 Nov 2016 09:03:59 +0000, Steve Pope wrote:
>
>> <mbjorn@y7mail.com> wrote:
>>
>>> A package which calls itself "an industry-leading scientific graphing
>>> and data analysis software"
>>> suggests breaking the samples into real and imaginary parts, and fitting
>>> curves to each. Hmmmph. I guess it is not a common task that they could
>>> be bothered coding.
Indeed, Origin is very cool for the sort of problems where you don't
have to write your own code.
>> Are the abscissae complex, or are they real?
>>
>> If real, I think doing this is the appropriate answer even if the range
>> is complex.
>
> Yes, at least if you're looking for a least-squares fit. If you're
> looking for a fit that minimizes some nonlinear cost criteria then the
> real and complex solutions may well interact.
>
If I get it correctly, in case of Gaussian noise the least-squares
criterion is the same thing as maximizing the likelihood function.
Incidentally, makes me feel a pity that while finding the global maximum
of the likelihood function is a standard method in communications, in
scientific packages fits work as some kinds of iterative processes, so
if you want to search for the global maximum (useful when the problem is
ill-defined) you have to write the code instead of just clicking a
couple of buttons in a standard package. ;)
Gene

0

Evgeny

11/14/2016 1:01:01 AM

On Mon, 14 Nov 2016 16:07:32 +0300, Evgeny Filatov wrote:
> On 14/11/2016 01:46, Tim Wescott wrote:
>> On Sat, 12 Nov 2016 09:03:59 +0000, Steve Pope wrote:
>>
>>> <mbjorn@y7mail.com> wrote:
>>>
>>>> A package which calls itself "an industry-leading scientific graphing
>>>> and data analysis software"
>>>> suggests breaking the samples into real and imaginary parts, and
>>>> fitting curves to each. Hmmmph. I guess it is not a common task that
>>>> they could be bothered coding.
>
> Indeed, Origin is very cool for the sort of problems where you don't
> have to write your own code.
>
>>> Are the abscissae complex, or are they real?
>>>
>>> If real, I think doing this is the appropriate answer even if the
>>> range is complex.
>>
>> Yes, at least if you're looking for a least-squares fit. If you're
>> looking for a fit that minimizes some nonlinear cost criteria then the
>> real and complex solutions may well interact.
>>
>>
> If I get it correctly, in case of Gaussian noise the least-squares
> criterion is the same thing as maximizing the likelihood function.
>
> Incidentally, makes me feel a pity that while finding the global maximum
> of the likelihood function is a standard method in communications, in
> scientific packages fits work as some kinds of iterative processes, so
> if you want to search for the global maximum (useful when the problem is
> ill-defined) you have to write the code instead of just clicking a
> couple of buttons in a standard package. ;)
That's because of the way the problem is structured. In comms, if the
noise isn't Gaussian then the maximum of the likelihood function is not
found in a least-squares fit. For my Master's thesis I worked on a
system that operated at 400kHz or so, where the noise primarily comes
from electrostatic discharge and where the best noise models all had
infinite variance. Under those circumstances, you basically threw out
all the tidy math that comes from the Gaussian assumption.
--
Tim Wescott
Control systems, embedded software and circuit design
I'm looking for work! See my website if you're interested
http://www.wescottdesign.com

0

Tim

11/14/2016 5:33:05 PM

On 14/11/2016 20:33, Tim Wescott wrote:
> That's because of the way the problem is structured. In comms, if the
> noise isn't Gaussian then the maximum of the likelihood function is not
> found in a least-squares fit. For my Master's thesis I worked on a
> system that operated at 400kHz or so, where the noise primarily comes
> from electrostatic discharge and where the best noise models all had
> infinite variance. Under those circumstances, you basically threw out
> all the tidy math that comes from the Gaussian assumption.
>
From the sound of it, looks like you might have used some erasure codes
like Reed-Solomon?
Gene

0

Evgeny

11/14/2016 6:17:39 PM

On Mon, 14 Nov 2016 21:17:39 +0300, Evgeny Filatov wrote:
> On 14/11/2016 20:33, Tim Wescott wrote:
>> That's because of the way the problem is structured. In comms, if the
>> noise isn't Gaussian then the maximum of the likelihood function is not
>> found in a least-squares fit. For my Master's thesis I worked on a
>> system that operated at 400kHz or so, where the noise primarily comes
>> from electrostatic discharge and where the best noise models all had
>> infinite variance. Under those circumstances, you basically threw out
>> all the tidy math that comes from the Gaussian assumption.
>>
>>
> From the sound of it, looks like you might have used some erasure codes
> like Reed-Solomon?
>
> Gene
I built a receiver that delivered bit-slice integrator levels to the
decoding algorithm. It was accompanied by a section in my thesis
explaining how those levels should be interpreted (IIRC, +/- 64 indicated
highest likelihood, with the likelihood dropping both above and below
that absolute value).
It was when the Coast Guard was first developing their differential GPS
service; my Thesis advisor had a bunch of graduate students who were
working on developing the codes.
--
Tim Wescott
Control systems, embedded software and circuit design
I'm looking for work! See my website if you're interested
http://www.wescottdesign.com

0

Tim

11/14/2016 6:33:31 PM

On 14.11.2016 21:33, Tim Wescott wrote:
> On Mon, 14 Nov 2016 21:17:39 +0300, Evgeny Filatov wrote:
>
>> On 14/11/2016 20:33, Tim Wescott wrote:
>>> That's because of the way the problem is structured. In comms, if the
>>> noise isn't Gaussian then the maximum of the likelihood function is not
>>> found in a least-squares fit. For my Master's thesis I worked on a
>>> system that operated at 400kHz or so, where the noise primarily comes
>>> from electrostatic discharge and where the best noise models all had
>>> infinite variance. Under those circumstances, you basically threw out
>>> all the tidy math that comes from the Gaussian assumption.
>>>
>>>
>> From the sound of it, looks like you might have used some erasure codes
>> like Reed-Solomon?
>>
>> Gene
>
> I built a receiver that delivered bit-slice integrator levels to the
> decoding algorithm. It was accompanied by a section in my thesis
> explaining how those levels should be interpreted (IIRC, +/- 64 indicated
> highest likelihood, with the likelihood dropping both above and below
> that absolute value).
>
> It was when the Coast Guard was first developing their differential GPS
> service; my Thesis advisor had a bunch of graduate students who were
> working on developing the codes.
>
Thanks; now I better appreciate your contribution.
I looked up Wikipedia on "differential GPS", and seen that "The United
States Coast Guard and Canadian Coast Guard each run such systems in the
U.S. and Canada on the longwave radio frequencies between 285 kHz and
325 kHz near major waterways and harbors." About the same frequencies
you've mentioned. Kinda cool!
Gene

0

Evgeny

11/14/2016 10:20:23 PM

On Tue, 15 Nov 2016 01:20:23 +0300, Evgeny Filatov wrote:
> On 14.11.2016 21:33, Tim Wescott wrote:
>> On Mon, 14 Nov 2016 21:17:39 +0300, Evgeny Filatov wrote:
>>
>>> On 14/11/2016 20:33, Tim Wescott wrote:
>>>> That's because of the way the problem is structured. In comms, if
>>>> the noise isn't Gaussian then the maximum of the likelihood function
>>>> is not found in a least-squares fit. For my Master's thesis I worked
>>>> on a system that operated at 400kHz or so, where the noise primarily
>>>> comes from electrostatic discharge and where the best noise models
>>>> all had infinite variance. Under those circumstances, you basically
>>>> threw out all the tidy math that comes from the Gaussian assumption.
>>>>
>>>>
>>> From the sound of it, looks like you might have used some erasure
>>> codes
>>> like Reed-Solomon?
>>>
>>> Gene
>>
>> I built a receiver that delivered bit-slice integrator levels to the
>> decoding algorithm. It was accompanied by a section in my thesis
>> explaining how those levels should be interpreted (IIRC, +/- 64
>> indicated highest likelihood, with the likelihood dropping both above
>> and below that absolute value).
>>
>> It was when the Coast Guard was first developing their differential GPS
>> service; my Thesis advisor had a bunch of graduate students who were
>> working on developing the codes.
>>
>>
> Thanks; now I better appreciate your contribution.
>
> I looked up Wikipedia on "differential GPS", and seen that "The United
> States Coast Guard and Canadian Coast Guard each run such systems in the
> U.S. and Canada on the longwave radio frequencies between 285 kHz and
> 325 kHz near major waterways and harbors." About the same frequencies
> you've mentioned. Kinda cool!
>
> Gene
Actually, I think those frequencies are correct and my memory was
faulty. I just fired up the receiver to check (my thesis advisor gave it
back to me about 10 years ago), but some of the LED segments are out --
but as far as I can tell that's the frequency range it supports.
The USCG transmitters were originally piggy-backed on their radio
direction finding beacons (I don't know if they even support that any
more); originally there was some thought that it would be nice to also be
capable of using aviation RDF beacons as well; those are at higher
frequencies, and are probably why I was remembering 400-ish instead of
300-ish.
According to my thesis advisor it was the second-ever design to work in
that service, which I take with a glow of pride and a grain of salt,
because I expect that SOMEONE must have been working on it.
(Radio direction finding, BTW, is the reason for those round antennas on
top of old airplanes: they were designed so that you'd get a null in
reception when the hole of the donut was pointed along a line to the
transmitter. Plot a few of those on a map, and you'd know where you
were.)
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
I'm looking for work -- see my website!

0

Tim

11/15/2016 1:01:01 AM

On 15/11/2016 03:01, Tim Wescott wrote:
>
> (Radio direction finding, BTW, is the reason for those round antennas on
> top of old airplanes: they were designed so that you'd get a null in
> reception when the hole of the donut was pointed along a line to the
> transmitter. Plot a few of those on a map, and you'd know where you
> were.)
>
Satellite navigation has by now evolved to centimeter-precision Real
Time Kinematic. I'm a complete ignoramus, but made me wondering whether
anyone has bothered to upgrade Radio direction finding with modern
techniques, such as spread spectrum for multipath mitigation or perhaps
more accurate beamforming with MIMO...
Gene

0

Evgeny

11/15/2016 12:05:36 PM

On Tue, 15 Nov 2016 15:05:36 +0300, Evgeny Filatov wrote:
> On 15/11/2016 03:01, Tim Wescott wrote:
>>
>> (Radio direction finding, BTW, is the reason for those round antennas
>> on top of old airplanes: they were designed so that you'd get a null in
>> reception when the hole of the donut was pointed along a line to the
>> transmitter. Plot a few of those on a map, and you'd know where you
>> were.)
>>
>>
> Satellite navigation has by now evolved to centimeter-precision Real
> Time Kinematic. I'm a complete ignoramus, but made me wondering whether
> anyone has bothered to upgrade Radio direction finding with modern
> techniques, such as spread spectrum for multipath mitigation or perhaps
> more accurate beamforming with MIMO...
>
> Gene
I don't pay much attention to that segment any more. The last I heard
there was still some debate about whether to scrap Loran or update it.
I'm pretty sure that RDF is inherently imprecise because it's noncoherent.
--
Tim Wescott
Control systems, embedded software and circuit design
I'm looking for work! See my website if you're interested
http://www.wescottdesign.com

0

Tim

11/15/2016 4:47:57 PM

On Tue, 15 Nov 2016 10:47:57 -0600, Tim Wescott <tim@seemywebsite.com>
wrote:
>On Tue, 15 Nov 2016 15:05:36 +0300, Evgeny Filatov wrote:
>
>> On 15/11/2016 03:01, Tim Wescott wrote:
>>>
>>> (Radio direction finding, BTW, is the reason for those round antennas
>>> on top of old airplanes: they were designed so that you'd get a null in
>>> reception when the hole of the donut was pointed along a line to the
>>> transmitter. Plot a few of those on a map, and you'd know where you
>>> were.)
>>>
>>>
>> Satellite navigation has by now evolved to centimeter-precision Real
>> Time Kinematic. I'm a complete ignoramus, but made me wondering whether
>> anyone has bothered to upgrade Radio direction finding with modern
>> techniques, such as spread spectrum for multipath mitigation or perhaps
>> more accurate beamforming with MIMO...
>>
>> Gene
>
>I don't pay much attention to that segment any more. The last I heard
>there was still some debate about whether to scrap Loran or update it.
>I'm pretty sure that RDF is inherently imprecise because it's noncoherent.
Even the VOR network will be partially dismantled, leaving just enough
as backup in case something happens to GPS/GLONASS. Very few
airplanes, even general aviation airplanes, even have ADF or NDB
equipment any more.
Aviation is weird. Some technologies get adapted quickly while other
critical parts of the system remain in the stone age.